They do work on Android. But of course they link to the web versions and not to any app.
It's just plain HTML links without any external dependencies which is the whole point of this plugin respecting privacy.
Messenger which is a deal killer.
You mean the Facebook messenger? I am not aware that it is possible to link to that via a plain url to invoke a new message. If you know about any plain url that can do this and it is, then it can be added.
Same for sharing to the FB app* (or others).
Or any option to invoke SMS.
It is not possible to invoke SMS via the web.
Addthis does not include Instagram either.
Instagram is very encapsulated within the app. It is not possible to share to an instagram accout via the web at all. You can only share within Instragram. To my knowledge.
Besides to my knowledge about statistics other make these sharing buttons are generally rarely used by anyone ;-)
*Edit: It seems it might be possible to to share to a FB app via an url but not sure if it requires any FB JS to be loaded. If it is not for this plugin.